SLIDE 1
HELSINGIN YLIOPISTO HELSINGFORS UNIVERSITET UNIVERSITY OF HELSINKI
Spatial dependence HELSINGIN YLIOPISTO HELSINGFORS UNIVERSITET - - PDF document
Spatial dependence HELSINGIN YLIOPISTO HELSINGFORS UNIVERSITET - - PDF document
Spatial dependence HELSINGIN YLIOPISTO HELSINGFORS UNIVERSITET UNIVERSITY OF HELSINKI Everything is related to everything else, but nearby things are more related than distant things Spatial Data Mining This is usually true even for spatially
SLIDE 2
SLIDE 3
Poisson process: from one to two dimensions
Easy to extend the Poisson process to a two-dimensional case Again, constant intensity λ The expected number of events in region A depends on the intensity and the area of A: E(X(A)) = λ|A| The spatial Poisson process is a model of what would happen if the events were independent from each other
No rst order variation No second order effects
First order variation: intensity
Instead of constant intensity λ an intensity function
λ(s) = lim
|ds|→0
E(X(ds))
|ds| ds a neighbourhood of point s E(X(ds)) the expected number of points in this neighbourhood
|ds| the size of the neighbourhood
The intensity at point s can be viewed as the density of events in an innitely small neighbourhood of s
Using the intensity function
A Poisson process can use the intensity function instead of a constant intensity Such a heterogeneous Poisson process models the rst order variation of a point pattern The expected number of events in a region A E(X(A )) =
- A
λ(s)ds
Estimating intensity
Kernel estimation Represent each point by a symmetrical two-dimensional density function, e.g. normal distribution Estimate the intensity function as the sum of these density functions
ˆ λτ(s) =
1
δτ(s)
n
- i=1
1
τ2 k s−si τ
- s1,...,sn event points
k kernel function
τ > 0 bandwidth δτ(s) edge correction
Kernel estimation
Bandwidth denes how far from each point the effect reaches In effect, it species how detailed the variation in intensity is
Simulating a Poisson process
Homogeneous Poisson process: two phases
- 1. Number of events in area A : n ∼ Poisson(λ|A |
- 2. The locations for the events can be obtained from
a uniform distribution over A
Similarly for a heterogeneous Poisson process
- 1. λ not constant
- 2. Locations from a non-uniform distribution
SLIDE 4
Measuring second order effects
Nearest neighbour measures
G(h): probability that the distance from a random event to the nearest other event ≤ h F(h): probability that the distance from a random location to the nearest event ≤ h
If events are clustered, G(h) < F(h) Only shows very small-scale attraction / repulsion Something else is required for scales larger than the nearest neighbour distance
K function
Measure for second order effects Basic case: constant λ, one point pattern
λK(h) = expected number of other events within
radius h of a random event For a homogeneous Poisson process K(h) = πh2
Also possible to measure Kinhom(h) for a heterogeneous point pattern For two point patterns
λjKij(h) = expected number of events of type j
within radius h of a random event of type i
K function: example
Two pairs of lake names
Mustalampi 'Black Pond' Valkealampi 'White Pond' Kuikkalampi 'Diver Pond' Ruunalampi 'Gelding Pond'
Spatial distributions and K functions
Blue line: homogeneous Kij Green line: heterogeneous Kinhom
ij
Modelling second order variation
Poisson cluster process Start with a Poisson process
Normally, a homogeneous process In principle, heterogeneous also possible, but difcult to estimate
This process generates parents Each parent generates a random number of daughters
Distributed independently around the parent These are the actual events
Spatially continuous phenomena
Observations from distinct points in space This time, measurements of a spatially continuous variable {Y(s),s ∈ R} Goal: model the behaviour of Y across R Again, useful to divide variation into rst and second order effects
First order properties of continuous data
Mean value surface {µ(s),s ∈ R},µ(s) = E(Y(s)) Normal statistical regression problem
Linear regression of Y(s) with spatial coordinates sx,sy Trend surface analysis More sophisticated methods available
Goal: interpolate the value of Y between the
- bservation points
Y(s) = µ(s)
SLIDE 5
Second order effects in continuous data
Usually better to assume Y(s) = µ(s)+U(s)
µ(s) global trend
U(s) spatially correlated residual, with
∀s ∈ R : E(U(s)) = 0
U(s) can be used to model second order effects Common assumption: U(s) is stationary
E(U(s)) and Var(U(s)) constant Cov(U(s),U(s′)) depends only on h = s′ −s In other words, the same in different parts of R
Often also isotropic
Cov(U(s),U(s′)) depends only on |h| In other words, the same in all directions
Predicting with second order effects
If the residual process {U(s),s ∈ R} is spatially correlated, it is possible to give better estimates than Y(s) = ˆ
µ(s)
Kriging: ˆ Y(s) = ˆ
µ(s)+ ˆ
U(s) Various methods for this
Beyond the scope of this course No general criterion for choosing, beyond see what works
Bottom line: modelling both rst and second order effects gives reasonably good predictions
Proximity in area data
Proximity matrix W wij =
- 1 if Ai and Aj share a border
0 otherwise
A B C D E F A 1 1 1 B 1 1 1 1 C 1 1 D 1 1 1 E 1 1 1 1 F 1 1 1 1
More elaborate measures for proximity possible
First order variation
Simple option: moving averages
Replace the value for each area by the averages of its neighbours
ˆ µi = n
j=1wijyj
n
j=1wij
Convert to point data
E.g. represent each area by its centre Perform kernel estimation
Median polish
For regular grids Represent each grid cell as yij = µ+ri +cj +εij ri, cj row and column trends, εij random error
Second order effects
Moran's I statistic: spatial correlation I = nn
i=1
n
j=1wij(yi − ¯
y)(yj − ¯ y)
n
i=1(yi − ¯
y)2
i=j wij
- Varies between −1 and +1, no autocorrelation
when I = 0
Geary's C statistic: variance of the difference of neighbouring values C =
(n−1)n
i=1
n
j=1wij(yi −yj)2
2
n
i=1(yi − ¯
y)2
i=j wij
- Varies between 0 and 2, no autocorrelation when